Factoring Method

Factoring a Quadratic

AC Method

Purpose:

   Factor the quadratic ax^2 + bx + c into this form: (      )(      ) of two terms multiplied.

Example:  

     2x^2 + 11x + 12

Method: 

     Find factors of a times c that when added equal b

Steps:

     given: a=2, b=11, c=12

     multiply a times c:  (2)(12)=24

     Factors of 24     Factors added

       1*24                   25

       2*12                   14

       3*8                     11

     Separate the example equation into the a times c factors that add to equal b:

       2x^2 + 11x + 12 = 2x^2 + 3x + 8x + 12

     Factor the first two terms and the last two terms

       2x^2 + 3x = x(2x +3) and 8x + 12 = 4(2x+3)

       

       x(2x+3) + 4(2x+3)

     Factor out the common term

       (2x + 3)(x + 4)

     The quadratic is now factored!

     If you cannot find two factors of a times c that add to equal b, the equation cannot be factored in 
     this way.

An example of when c is negative:

     2x^2 + 5x - 12

     a=2, b=5, c=-12

     (a)(c) = (2)(-12)= -24

     Factors of 24 that subtract to equal b:

     1*24      23

     2*12      10

     3*8          5

     2x^2 - 3x + 8x - 12

     x(2x - 3) + 4(2x - 3)

     (2x - 3)(x + 4) the answer

An example of when b is negative:

     3x^2 -4x -4

     a=3, b=-4, c=-4

     (a)(c) = (3)(-4) = -12

     Factors of 12 that subtract to equal b:

     1*12       11

     2*6           4

     3x^2 + 2x - 6x -4

     x(3x + 2) - 2(3x + 2)

     (3x + 2)(x - 2) the answer

An example of when it doesn't work:

     3x^2 + 11x + 1

     a=3, b=11, c=1

     (a)(c) = (3)(1) = 3

     Factors of 3 are:

     1*3   added equals 4, since b does not equal 4, this equation cannot be factored this way.